1. Introduction and Preliminaries

where the parameters Open image in new window and Open image in new window are positive numbers and initial conditions Open image in new window and Open image in new window are arbitrary numbers. System (1.1) was mentioned in [1] as a part of Open Problem 3 which asked for a description of global dynamics of three specific competitive systems. According to the labeling in [1], system (1.1) is called Open image in new window. In this paper, we provide the precise description of global dynamics of system (1.1). We show that system (1.1) has a variety of dynamics that depend on the value of parameters. We show that system (1.1) may have between zero and two equilibrium points, which may have different local character. If system (1.1) has one equilibrium point, then this point is either locally saddle point or non-hyperbolic. If system (1.1) has two equilibrium points, then the pair of points is the pair of a saddle point and a sink. The major problem is determining the basins of attraction of different equilibrium points. System (1.1) gives an example of semistable non-hyperbolic equilibrium point. The typical results are Theorems 4.1 and 4.5 below.

System (1.1) is a competitive system, and our results are based on recent results developed for competitive systems in the plane; see [2, 3]. In the next section, we present some general results about competitive systems in the plane. The third section deals with some basic facts such as the non-existence of period-two solution of system (1.1). The fourth section analyzes local stability which is fairly complicated for this system. Finally, the fifth section gives global dynamics for all values of parameters.

Competitive systems were studied by many authors; see [4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19], and others. All known results, with the exception of [4, 6, 10], deal with hyperbolic dynamics. The results presented here are results that hold in both the hyperbolic and the non-hyperbolic cases.

We now state three results for competitive maps in the plane. The following definition is from [18].

The following theorem was proved by de Mottoni and Schiaffino [20] for the Poincaré map of a periodic competitive Lotka-Volterra system of differential equations. Smith generalized the proof to competitive and cooperative maps [15, 16].

The following theorems were proved by Kulenović and Merino [3] for competitive systems in the plane, when one of the eigenvalues of the linearized system at an equilibrium (hyperbolic or non-hyperbolic) is by absolute value smaller than Open image in new window while the other has an arbitrary value. These results are useful for determining basins of attraction of fixed points of competitive maps.

Our first result gives conditions for the existence of a global invariant curve through a fixed point (hyperbolic or not) of a competitive map that is differentiable in a neighborhood of the fixed point, when at least one of two nonzero eigenvalues of the Jacobian matrix of the map at the fixed point has absolute value less than one. A region Open image in new window is rectangular if it is the Cartesian product of two intervals in Open image in new window.

For maps that are strongly competitive near the fixed point, hypothesis b. of Theorem 1.5 reduces just to Open image in new window. This follows from a change of variables [18] that allows the Perron-Frobenius Theorem to be applied to give that, at any point, the Jacobian matrix of a strongly competitive map has two real and distinct eigenvalues, the larger one in absolute value being positive, and that corresponding eigenvectors may be chosen to point in the direction of the second and first quadrants, respectively. Also, one can show that in such case no associated eigenvector is aligned with a coordinate axis.

The following result gives a description of the global stable and unstable manifolds of a saddle point of a competitive map. The result is the modification of Theorem 1.7 from [12].

If the conditions of this theorem are satisfied, then (2.6) implies that there is no real (if the first condition of this theorem is satisfied) or positive equilibrium points (if the second condition of this theorem is satisfied).

Kulenović MRS, Merino O: Invariant manifolds for competitive discrete systems in the plane. to appear in International Journal of Bifurcation and Chaos, http://arxiv.org/abs/0905.1772v1 to appear in International Journal of Bifurcation and Chaos,

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